The Riemannian curvature identities for the torsion connection on Spin ( 7 ) ${\rm Spin}(7)$ —Manifold and generalized Ricci solitons

It is shown that on compact $\mathrm{Spin}\left(7\right)$-manifold with exterior derivative of the Lee form lying in the Lie algebra $\mathrm{Spin}\left(7\right)$ the curvature $R$ of the $\mathrm{Spin}\left(7\right)$–torsion connection $R\in S^2{\Lambda }^2$ with vanishing Ricci tensor if and only if the 3-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the 3-form torsion is parallel with respect to the Levi-Civita and to the $\mathrm{Spin}\left(7\right)$-torsion connections simultaneously. Precise conditions for a compact $\mathrm{Spin}\left(7\right)$-manifold to has closed torsion are given in terms of the Ricci tensor of the $\mathrm{Spin}\left(7\right)$-torsion connection. It is shown that a compact $\mathrm{Spin}\left(7\right)$-manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $\mathrm{Spin}\left(7\right)$-manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $\mathrm{Spin}\left(7\right)$-structure.